$f(x, y, z) = y - z^2 - \sin(x)$ What are all the critical points of $f$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\left( \dfrac{\pi}{2} + \pi k, 0, 0 \right)$, where $k = \ldots -1, 0, 1 \ldots$ (Choice B) B $\left( \dfrac{\pi}{2} + \pi k, 0, -1 \right)$, where $k = \ldots -1, 0, 1 \ldots$ (Choice C) C $\left( \pi k, 0, -1 \right)$, where $k = \ldots -1, 0, 1 \ldots$ (Choice D) D There are no critical points.
Answer: A critical point of a scalar field $f$ is where $\nabla f = \bold{0}$. [What's that bolded 0?] Let's find the gradient of $f$ ! $\nabla f = \begin{bmatrix} -\cos(x) \\ \\ 1 \\ \\ -2z \end{bmatrix}$ Notice that the $y$ -component of the gradient is always $1$. In other words, no matter what input we feed the gradient of $f$, it will never equal the zero vector. Therefore, there are no critical points.